Amini, Massoud; Essmaili, Morteza; Filali, Mahmoud The second transpose of a derivation and weak amenability of the second dual Banach algebras. (English) Zbl 1354.46049 New York J. Math. 22, 265-275 (2016). Summary: Let \(\mathcal A\) be a Banach algebra, \(\mathcal A^*\), \(\mathcal A^{**}\) and \(\mathcal A^{***}\) be its first, second and third dual, respectively. Let \(R:\mathcal A^{***} \to \mathcal A^*\) be the restriction map, \(J: \mathcal A^* \to \mathcal A^{***}\) be the canonical injection and \(\Lambda: \mathcal A^{***} \to \mathcal A^{***}\) be the composition of \(R\) and \(J\). Let \(D:\mathcal A \to \mathcal A^*\) be a continuous derivation and \(D^{\prime \prime}: \mathcal A^{**} \to \mathcal A^{***}\) be its second transpose. We obtain a necessary and sufficient condition for \(\Lambda \circ D^{\prime \prime}: \mathcal A^{**} \to (\mathcal A^{**})^*\) to be a derivation. We apply this to prove some results on weak amenability of second dual Banach algebras. Cited in 1 ReviewCited in 2 Documents MSC: 46H20 Structure, classification of topological algebras 47B47 Commutators, derivations, elementary operators, etc. Keywords:derivation; second dual of Banach algebras; weak amenability PDFBibTeX XMLCite \textit{M. Amini} et al., New York J. Math. 22, 265--275 (2016; Zbl 1354.46049) Full Text: EMIS